Also, in [1]} it is obtained that the solution map is continuous dependence while not uniformly continuous dependence on initial data for the Burgers equation (REF ) in the same space \(H^s(\mathbb {R})\)  with \(s>3/2\) . For the endpoint case, Linares et al. in [2]} proved that the Cauchy problem for (REF ) is ill-posed in \(H^{3/2}(\mathbb {R})\) , where the key point is that the available local well-posedness theory in \(H^s(\mathbb {R})\)  with any \(s>3/2\)  have be used, for more details see Remark 1.6. Using the idea developed in [2]}, Guo et al. in [4]} to prove the ill-posedness for the Camassa-Holm equation in the critical Sobolev space \(H^{3/2}(\mathbb {R})\)  and even in the Besov space \(B^{1+1/p}_{p,r}(\mathbb {R})\)  with \(r>1\) . Precisely speaking, their main idea is to construct a blow-up smooth solution \(u(t)\in \mathcal {C}([0,T^*),B^{1+1/p}_{p,r})\cap \mathcal {C}([0,T^*),H^2)\)  such that
\(\Vert u_0\Vert _{B^{1+\frac{1}{p}}_{p,r}}\le \varepsilon \quad \text{and} \quad \lim _{t\uparrow T^*}\Vert u(t)\Vert _{\mathrm {Lip}}=\infty ,\quad T^*<\varepsilon .\) 
